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Cross-posted to Math Educators Stack Exchange. (link)

I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. I am interested in the best materials available in English, French, German or Hebrew.

Ideally, the book(s) should provide a comprehensive introduction to algebra at this level, starting from the most basic operations on polynomials. It should include necessary theory (e.g., Bezout's remainder theorem on polynomials, proof of the fundamental theorem of arithmetic, Euclid's algorithm, a more honest discussion of real numbers than usual, proofs of the properties of rational exponents, etc., and a general attitude that all statements are to be proved, with few exceptions). It should also have problems that range from exercises acquainting students with the basic algebraic manipulations on polynomials to much more difficult ones.

Specifically, I am looking for something similar in spirit to a series of excellent Russian books by Vilenkin for students in so-called "mathematical schools" from grades 8 to 11, although I am only looking for the equivalent of the grade 8 and 9 books, which are at precalculus level. To give you an idea, here are a sample of typical problems from the grade-8 book.

  1. Perform the indicated operations. $\frac{3p^2mq}{2a^2 b^2} \cdot \frac{3abc}{8x^2 y^2} : \frac{9a^2 b^2 c^3}{28pxy}$

  2. Prove that when $a \ne 0$, the polynomial $x^{2n} + a^{2n}$ is divisible neither by $x + a$ nor by $x - a$.

  3. Prove that if $a + b + c = 0$, then $a^3 + b^3 + c^3 + 3(a + b)(a + c) (b + c) = 0$.

  4. Prove that if $a > 1$, then $a^4 + 4$ is a composite number.

  5. Prove that if $n$ is relatively prime to $6$, then $n^2 - 1$ is divisible by 24.

  6. Simplify $\sqrt{36x^2}$.

  7. Simplify $\sqrt{12 + \sqrt{63}}$.

  8. Prove that the difference of the roots of the equation $5x^2 -2(5a + 3)x + 5a^2 + 6a + 1 = 0$ does not depend on $a$.

  9. Solve the inequality $|x - 6| \leq |x^2 - 5x + 2|$.

And here are the chapter titles for the grade 8 and 9 books.

Grade 8: Fractions. Polynomials. Divisibility; prime and composite numbers. Real numbers. Quadratic equations; systems of nonlinear equations; resolution of inequalities.

Grade 9: Elements of set theory. Functions. Powers and roots. Equations and inequalities, and systems thereof. Sequences. Elements of trigonometry. Elements of combinatorics and probability theory.

Broadly similar questions have been asked elsewhere, however the suggestions made there are not satisfactory for my purposes.

  1. The English translations of Gelfand's books are good; however they are not a sufficiently broad introduction to high school algebra, and do not have enough material on computational technique. They are more in the nature of supplements to an ordinary textbook.

  2. Some 19th century books like Hall and Knight have been suggested. On conceptual material, these tend to be too old in language and outlook.

  3. Basic Mathematics by Serge Lang seems more to dabble in various topics than to provide a thorough introduction to algebra.

  4. I am not inclined towards books with a very strong "New Math" orientation (1971-1983 France, for example). I don't think a student should need to understand the group of affine transformations of $\mathbb{R}$ to know what a line is.

Also, previous questions have perhaps focused implicitly on material in English. I have in mind a student who can also easily read French, German or Hebrew if something better can be found in those languages.

Edit. I'd like to clarify that I'm not asking for something identical to these books, just something as close as possible to their spirit. Fundamentally, this means: 1. It is a substitute for, rather than just a complement to, a regular school algebra textbook. 2. It is directed at the most able students. 3. It conveys the message that proofs and creative problem-solving are central to mathematics.

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    $\begingroup$ There is also a math educator forum on stack exchange where you may reach a more targeted audience. matheducators.stackexchange.com $\endgroup$ – citronrose Jun 1 '15 at 10:16
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    $\begingroup$ @citronrose Is it frowned on to post the same question on both forums? $\endgroup$ – Keith Jun 1 '15 at 10:26
  • $\begingroup$ If after some time you don't get satisfying answers here, you can maybe just move your question to the other forum. See meta.math.stackexchange.com/q/5028 $\endgroup$ – citronrose Jun 1 '15 at 10:56
  • $\begingroup$ Discourse on Algebra by Shafarevich doesn't have all that you want, but it could be useful. The various chapters were previously published in the journal "The Teaching of Mathematics", and they can be found freely available on the internet by googling "Shafarevich" along with the phrase "Selected Chapters from Algebra". $\endgroup$ – Dave L. Renfro Jun 1 '15 at 15:42
  • $\begingroup$ @DaveL.Renfro Thanks for pointing that out. $\endgroup$ – Keith Jun 3 '15 at 3:56
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Here is my second try. I give some references for Olympiad-style problem solving. Hopefully you'll find something useful in each of them.

As a final warning, I must tell you that (at least in my own experience) contest-geared textbooks tend to focus in quick development of problem-solving skills rather than rigorous mathematical exposition. You may want to consider other kind of textbook to compensate for this.

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I would suggest you to take a look at the book Mathematical Thinking: Problem-Solving and Proofs by John P. D'Angelo and Douglas B. West. It covers a broad array of undergraduate-level topics in a self-contained way and starting from basic notions that are likely to be familiar to the kind of student you describe (mathematical logic, methods of proof, sets and functions). The authors present a blend of rigorous theoretical exposition with a practical problem-solving approach through lots of exercises. However a disadvantage is that the scope of some topics is rather limited (most likely due to constraints of space, but I think this is inevitable given the number of subjects covered). Hope you find it useful!

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  • $\begingroup$ I don't think that book is suitable. It clearly has proficiency in high-school algebra as a prerequisite. The idea is that a student would learn proofs and problem-solving through a different kind of presentation of normal high-school topics, primarily in algebra. $\endgroup$ – Keith Jun 9 '15 at 21:39
  • $\begingroup$ Please excuse me for misunderstanding your request. It seems to me that mathematical contests may satisfy the needs you just described: it would allow you to introduce standard high school material via unusual and challenging problems. (In passing, I must say that the sample questions in your first post are reminiscent of olympiad-style problems). Perhaps you'd be interested in some references? $\endgroup$ – Gabriel Chicas Reyes Jun 9 '15 at 23:07
  • $\begingroup$ Sure, if you know some good books, I'd appreciate it. Whatever comes closest to the criteria, although being a complete substitute for an algebra textbook would seem to be out of the question for that kind of book. $\endgroup$ – Keith Jun 9 '15 at 23:17
  • $\begingroup$ Alright, I will try to answer again. $\endgroup$ – Gabriel Chicas Reyes Jun 10 '15 at 0:29
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I just saw this today, two years later. There is an organization and website geared specifically to these types of young students. It's called, "Art of Problem Solving." That is the name of the web site. The banner page states, "Is math class too easy for you? You've come to the right place."

They have textbooks, videos, online courses, competition preparation, etc.

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